Global and Non-Global Solutions for Pseudo-Parabolic Equation with Singular Potential

Abstract

This paper considers initial boundary value to a pseudo-parabolic equation with singular potential $\frac{u_{t}}{|x|^{s}}-\Delta u_{t}-\Delta u=|u|^{p-2} u$ with $2<p<\frac{2 N}{N-2}$, which was studied in [1] by Lian et al. They dealt with the global existence, asymptotic behavior with low initial level $J\left(u_{0}\right) \leq d$ and got the blow-up conditions of solutions with low and high initial level. In this paper, we give a new blow-up result which independent of the initial Nehari functional $I\left(u_{0}\right)$, and estimate the lower bound for blow-up time under some conditions. Finally, the precise exponential decay estimate is obtained for global solution with some conditions.

Keywords

Pseudo-parabolic equation / singular potential / blow-up / bounds for blow up time / exponential decay